ERIC ED462272: Geometry and Algebra: The Future Flight Equation. A Lesson Guide with Activities in Mathematics, Science, and Technology. NASA CONNECT.

27 Pages·2001·0.77 MB·English

by  ERIC

DOCUMENT RESUME
ED 462 272 SE 065 382
Geometry and Algebra: The Future Flight Equation. A Lesson
TITLE
Guide with Activities in Mathematics, Science, and
Technology. NASA CONNECT.
National Aeronautics and Space Administration, Hampton, VA.
INSTITUTION
Langley Research Center.
REPORT NO EG-2001-09-30-LARC
2001-00-00
PUB DATE
30p.; Accompanying videotape not available from ERIC.
NOTE
AVAILABLE FROM
For full text: http://spacelink.nasa.gov/products.
Classroom
Guides
Teacher (052)
PUB TYPE
EDRS PRICE
MF01/PCO2 Plus Postage.
*Aerospace Education; Algebra; Aviation Mechanics; Aviation
DESCRIPTORS
Technology; *Design; Geometry; *Integrated Activities;
Learning Activities; *Mathematics Activities; Mathematics
Education; *Science Activities; Science Education; Secondary
Education; Technology Education
ABSTRACT
This activity, part of the NASA CONNECT Series, is designed
to help students in grades 6-8 learn how NASA engineers develop experimental
aircraft. It consists of an overview of the program, details of the hands-on
activity, a series of blackline master student worksheets, teacher materials,
and a guide to further resources.
(MM)
Reproductions supplied by EDRS are the best that can be made
from the original document.
0
Educators
Grades 6-8
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U S DEPARTMENT OF EDUCATION
Office of Educational Research and Improvement
EDUCATIONAL RESOURCES INFORMATION
Ice,CENTER (ERIC)
Th s document has been reproduced as
ceived from the person or organization
originating it
0 Minor changes have been made to
improve reproduction quality
°
Points of view or opinions stated in this
document do not necessarily represent
official OERI position or policy
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Geometry and Algebra:The Future Flight Equation is
available in electronic format through NASA
Space link one of NASA's electronic resources
specifically developed for the educational
community.This publication and other educational
products may be accessed at the following
address: http://spacelink.nasa.gov/products
A PDF version of the lesson guide for NASA
CONNECT can be found at the NASA CONNECT
web site: http://connect.larc.nasa.gov
3
Geometry and Algebra:
The Future Flight Equation
A Lesson Guide with Activities in Mathematics, Science, and Technology
Student Worksheets
Program Overview
Data Chart
Summary and Objectives
13
5
Wing Chart
14
Student Involvement
5
Experimental Data and Wing Charts
15
Cue Card Questions
5
Delta Wing Template
16
Hands-On Activity
5
Oblique Wing/Stabilizer Template
17
Instructional Technology Activity
5
Straight Wing/Stabilizer Template
18
Resources
5
Swept-Back Wing Template
19
Hands-On Activity
20
Graph Paper
Background
6
Fuselage Template
21
National Standards
22
Cue Cards
6
Instructional Objectives
7
Teacher Materials
8
Vocabulary
Cue Card Answers
23
Preparing for the Activity
8
Student Materials
8
Instructional Technology Activity
Teacher Materials
8
24
Description
Time
8
24
National Standards
8
Focus Questions
Instructional Objectives
25
Advance Preparation
8
The Activity
9
Resources
Extensions
12
26
Books, Pamphlets, and Periodicals
26
Web Sites
Acknowledgments: Special thanks to Summer 2001 Educators in Residence,Jennifer Pulley, Chris Giersch, Bill Williams,
and NCI-M.
NASA CONNECT is a production of the NASA Langley Research Center, Hampton, VA. All Rights Reserved. Broadcast and off-air rights are unlimited and are
granted in perpetuity with the following stipulations:NASA CONNECT shall not be used for commercial purposes; used, in whole or in part, to endorse a
commercial product; stored, in whole or in part, in a commercial database; altered electronically, mechanically, or photographically without the expressed
and prior written permission of NASA.This publication is in the public domain and is not protected by copyright. Permission is not required for duplication.
2001-2002 NASA CONNECT Series
Program Overviow
Flight Equation.They will observe NASA engineers
In Geometry and Algebra:The Future Flight Equation,
using geometry and algebra when they measure
students will learn how NASA engineers develop
and design models to be tested in wind tunnels. By
experimental aircraft. They will learn about the
conducting hands-on and web activities, students
Hyper-X Research Vehicle, an experimental plane
will make connections between NASA research and
that uses scramjet engine technology to propel
the mathematics, science, and technology they learn
itself to ten times the speed of sound. Students will
in their classrooms.
understand how the Hyper-X is part of the Future
6
standards. Students are designated as Aeronautical
Cue Card Questions
Engineers in Training (AET).They will analyze wing
Norbert, NASA CONNECT's animated cohost, poses
geometry based on measurements and
questions throughout the broadcast.These
observations. Students will use geometry and
questions direct the instruction and
algebra to design, construct, and test an
encourage students to think about the
experimental wing.
concepts being presented. When viewing a
videotaped version of NASA CONNECT,
Instructional Technology Activity
educators have the option to use Norbert's
PlaneMath, the instructional technology activity, is
Pause, which gives students an opportunity
aligned with the National Council of Teachers of
to reflect and record their answers on the Cue
Mathematics (NCTM) standards, the National
Cards (p.22). Norbert appears with a remote to
Science Education (NSE) standards, the International
indicate an appropriate time to pause the videotape
Technology Education Association (ITEA) standards,
and discuss the answers to the questions.
and the National Educational Technology (NET)
standards.This online interactive activity lets
Hands-On Activity
students learn, design, and test experimental
The hands-on activity is teacher-created and is
aircraft.The students will learn about the forces of
aligned with the National Council of Teachers of
flight, wing shape, propulsion, experimental design,
Mathematics (NCTM) standards, the National
and several other topics.To access PlaneMath, go to
Science Education (NSE) standards, the International
Dan's Domain on NASA CONNECT'S web site at
Technology Education Association (ITEA) standards,
http://connect.larc.nasa.gov/dansdomain.html.
and the National Educational Technology (NET)
ki7
http://connect.larc.nasa.gov, offers online
Teacher and student resources (p.26) support,
resources for teachers, students, and parents.
enhance, and extend the NASA CONNECT program.
Teachers who would like to get the most from the
Books, periodicals, pamphlets, and web sites provide
NASA CONNECT web site can visit the Lab Manager,
teachers and students with background information
located in Dan's Domain,
and extensions. In addition to the resources listed in
http://connect.larc.nasa.gov/dansdomain.html.
this lesson guide, the NASA CONNECT web site,
Geometry and Algebra: The Future Flight Equation
EG-2001-09-30-LARC
2001-2002 NASA CONNECT Series
Hands-On Activity
1
Aeronautical research usually begins with
wings. They fly with jet engines, rocket engines,
computers, wind tunnels, and flight simulators, but
piston engines, solar-electric engines, and even
eventually the theories must fly. That is when flight
without engines. Some research planes are too
research begins, and aircraft are the primary tools of
small for a pilot; some are as large as an airliner.
the trade. There are four stages in the development
The first experimental planes designed exclusively
of new aircraft: mission (purpose), design
for research were the XS-1 and the D-558-1. They
(aerodynamics, propulsion, stability, control),
were made in 1946 to enable scientists to study
computer modeling, and testing.
flight near the speed of sound. Custom-made
Flight research involves doing precision maneuvers
planes were the only way to accomplish this
in either specially built experimental aircraft or in an
objective because supersonic wind tunnels were not
existing production aircraft that has been modified.
accurate enough, and no other planes had flown
All research aircraft are able to perform scientific
that fast. The supersonic era began when the XS-1
experiments because of the onboard instruments
broke the "sound barrier" in 1947.
that record data about its systems, aerodynamics,
In the 1950s the famous "X-Planes"continued to
and the outside environment.
take people to higher altitudes and greater speeds.
NASA pilots work closely with engineers to conduct
They were the first aircraft to fly at Mach 2 and Mach
carefully constructed flight programs that gradually
3, and the studies performed with them influenced
probe an aircraft's capability; edging toward the
the designs of all supersonic planes.
speed, altitude, and structural limits that will define
In the 1960s, the X-15 rocket plane became the first
the final performance of an aircraft. This procedure
aircraft to fly into space. The projects done on this
furnishes answers that will verify, extend, and
aircraft benefited not only NASA's Apollo Lunar
perhaps correct the inputs from computer studies,
Landing Program, but also the Space Shuttle, nearly
wind tunnel tests, and simulations. It is the last step
15 years later.
in the developmental process and leads the way for
designs that can be put into production. It also
Since the 1970s, NASA flight research has become
delivers the final word on a most crucial question:
more comprehensive with flights involving everything
How well does it fly?
from Space Shuttles to ultralight aircraft. NASA now
flies not only the fastest airplanes, but some of the
Experimental research aircraft are tools of
slowest as well. Flying machines continue to evolve
exploration, incorporating the newest ideas in every
with new wing designs, propulsion systems, and
aspect of aerospace flight. For this reason they come
flight controls. As always, a look at today's
in many shapes and sizes. They have short wings,
experimental research aircraft is a preview of the
delta wings, swept wings, moveable wings, and no
future.
141034)
Mathematics (NCTM) Standards
Develop and evaluate inferences that are based on
data
Compute fluently and make reasonable estimates
Build new mathematical knowledge through
Understand measurable attributes of objects and
problem solving
the units, systems, and processes of measurement
Apply and adapt a variety of appropriate strategies
Apply appropriate techniques, tools, and formulas
to solve problems
to determine measurements
Geometry and Algebra: The Future Flight Equation
EG-2001-09-30-LARC
2001-2002 NASA CONNECT Series
Physical Science
Recognize, use, and learn about mathematics in
contexts outside of mathematics
Motions and forces
Analyze characteristics and properties of two- and
Science and Technology
three-dimensional geometric shapes and develop
Abilities of technological design
mathematical arguments about geometric
Understanding about science and technology
relationships
Use visualization, spatial reasoning, and geometric
Technology (ITEA) Standards
modeling to solve problems
The Nature of Technology
Understand and use metric systems of
Develop an understanding of the characteristics
measurement
and scope of technology
Formulate questions that can be addressed with
Develop an understanding of the core concepts of
data and collect, organize, and display relevant
technology
data to answer them
Design
Solve problems that arise in mathematics and in
Develop an understanding of the attributes of
other contexts
design
Monitor and reflect on the process of
Develop an understanding of engineering design
mathematical problem solving
Develop an understanding of the role of
Make and investigate mathematical conjectures
troubleshooting, research and development,
Select and use various types of reasoning and
invention and innovation, and experimentation in
methods of proof
problem solving
Organize and consolidate mathematical thinking
through communication
Technology (NET) Standards
Recognize and use connections among
Technology tools to enhance learning, increase
mathematical ideas
learning, and promote creativity
Understand how mathematical ideas interconnect
Use technology resources for solving problems and
and build on one another to produce a coherent
making informed decisions
whole
Select and use appropriate tools and technology
resources to accomplish a variety of tasks and
Science (NSE) Standards
solve problems
Abilities necessary to do scientific inquiry
Understanding about scientific inquiry
The student will
design, construct, and test an experimental wing to
use algebra to calculate wing area, wingspan,
achieve maximum distance.
chord length, and aspect ratio.
incorporate collaborative problem-solving
use a portable glider catapult to analyze wing
strategies in a real-life application.
geometry based on measurement (distance rating)
and observations (glide rating and speed rating).
7
4.
Geometry and Algebra: The Future Flight Equation
EG-2001-09-30-LARC
2001-2002 NASA CONNECT Series
lift
aspect ratio - wingspan length divided by the
a force that is perpendicular to the air flow
average chord length
around the aircraft
thrust
chord straight line distance joining the leading
a force created by the engines that pushes
and trailing edge of an airfoil
an aircraft through the air
fuselage the part of the airplane to which the tail
weight the force of gravity acting on an object.
and wings are attached.The fuselage holds
The weight force pulls an aircraft toward the Earth
passengers and cargo and is streamlined to produce
and must be overcome by a combination of lift and
the least possible drag.
thrust.
horizontal stabilizer the horizontal part of the
wingspan distance from wing tip to wing tip
tail.The horizontal stabilizer helps to increase the
stability of the aircraft.
I
/I
'
Student Materials (per 4-student group)
Advance Preparation
20 straight pins
calculator
The teacher should construct a Portable Glider
4 small binder clips
masking tape
Catapult (PGC) for each student group.
Student Worksheets (p.13
21)
A. Cut out a 28-cm by 40-cm piece of cardboard.
10 meat trays (suggestion:28 cm x 23 cm size 12)
meter stick or measuring tape
B. Place two wooden rulers or paint sticks on top of
the cardboard and attach them to the board
4 scissors (optional: plastic knife, box cutter)
fine sand paper or emery board
with small binder clips, as shown in Figure 1.
Thumbtacks
Teacher Materials
4 small binder clips
2 thumbtacks
'
rubber band (size 64)
cardboard (28 cm x 40 cm)
t
..'
2 wooden rulers or paint sticks
:
..
Rubber band
:
*. 26 cm
Rubber band
Time
in launcho.:
:
Discussion of the activity
5 min
position
Preparing aircraft
25 min
:
Conducting the activity
60 min
..
Focus Questions
1. What are some common geometric shapes used
in wing design?
2. Why is the geometric shape of a wing important
Launch position
in aircraft design?
3. What are some tools engineers need to design
aircraft?
4. Why is it essential for engineers to work in teams
C. Cut a size 64 rubber band. Attach the rubber
when designing aircraft?
band to each of the rulers with thumbtacks as
shown in Figure 1. Fold (double) the ends of the
rubber band once before pinning the thumbtack
8
Geometry and Algebra: The Future Flight Equation
EG-2001-09-30-LARC
2001-2002 NASA CONNECT Series
Triangle
through it to prevent the rubber band from
A = (112)bh
tearing at the thumbtack.
b = base, h = height
D. Construct a horizontal line 2 6 cm from the top.
This line is called the launch position and
b
indicates how far back you pull the rubber band.
b,
Trapezoid
The following area formulas will be used to calculate
A = (1/2)(bi + b2)h
the wing area:
bi= base 1, b2= base 2
h = height
Rectangle
b,
A = lw
I = length, w = width
Ellipse
A = nab
it
= 3.14
a = semimajor axis
b = semiminor axis
Step 2: Conducting the Activity
- Part I: Preflight
Step 1: Introducing the Activity
A. Have students calculate the wing area (in cm2)
A. Announce: NASA has designated the class as
for each wing. Students have two options for
Aeronautical Engineers in Training (AET).Your
calculating wing area:1. the teacher may provide
job is to test current wing designs based on
formulas found in the lesson guide (students
distance traveled, glide rating, and speed rating.
may need to divide wings into shapes to
From your analysis of the data that you collect,
calculate the area) 2. students can count the
you will have the task of designing and testing
number of squares on the wing templates. If
an experimental wing to achieve maximum
students count the number of squares on the
distance traveled.
wing templates, they will have to estimate the
B. Organize students into groups of four.
number of squares around the edges of the
C. Distribute a Portable Glider Catapult to each
wings. Record the wing area on the data chart.
group along with the necessary materials.
B. Have students calculate the wingspan (in cm) for
D. Have students cut out the templates for the
each wing. The wingspan is the linear distance
fuselage, wings, and horizontal stabilizers.
from wing tip to wing tip. See Figure 2. Record
E. Have students place the templates on the meat
the value on the data chart.
trays and trace around the templates. Each
group should make four different fuselages and
\
f
one set of wings for each fuselage. Students
wingspan
should make a fifth fuselage to be used in
root chord
conjunction with their experimental wing (Part
tip chord
III of the activity).
I
k
To avoid wasting meat trays, we suggest students
follow the guideline:
Figure 2
One meat tray will yield two fuselages.
One meat tray will yield one fuselage and three
C. Have students measure the root chord, the width
horizontal stabilizers.
of the wing at the line of intersection with the
Use one meat tray for each of the wings.
fuselage, for each wing. See Figure 2. Record the
value on the data chart.
F. Have students tape a piece of masking tape to
D. Have students measure the tip chord, the width
the nose of the fuselage to prevent it from breaking.
Geometry and Algebra: The Future Flight Equation
EG-2001-09-30-LARC
9
2001-2002 NASA CONNECT Series
of the tip of the wing, for each wing. See Figure
Students should adhere to the following
2. Record the value on the data chart.
guidelines for corresponding wing shape and
horizontal stabilizer:
Note:The measure of the tip chord for a Delta Wing
Oblique wing
Oblique horizontal stabilizer
and an Oblique Wing is 0.
Delta wing
No horizontal stabilizer
E. Have students calculate the average chord by
Straight wing Straight horizontal stabilizer
using the formula: (root chord + tip chord) / 2.
Swept-back wing
Swept-back horizontal
Record the value on the data chart.
stabilizer
F. Have students calculate the aspect ratio for each
Have students construct each aircraft by placing
wing by using the formula: (wingspan) /
the corresponding wing and horizontal stabilizer
(average chord). Record the value on the data
into the fuselage. Have students secure each
chart.
wing and horizontal stabilizer with straight pins.
G. Have students use masking tape to mark the
See Figure 4.
launching point for each team. From the
launching point, have students put down a piece
of masking tape 12 meters long. Have students
measure and mark the tape at 1-meter intervals.
See Figure 3.
Oblique wing
Straight wing
12 m
Swept-back wing
Delta wing
= straight pin
Figure 4
0 Note:Two straight pins for the wing and one straight
pin for the horizontal stabilizer are sufficient to secure
the aircraft.
Desk
()Caution: Remind students
to use straight pins safely.
PGC
Step 3: Conducting the Activity - Part II:Test
Flight
A. In each group, one student should be in charge
of launching the aircraft, one student in charge
Figure 3
of stabilizing the Portable Glider Catapult, one
student in charge of marking where the aircraft
H. Have students place a desk or table at the
hit the ground, and one student in charge of
launching line to elevate the PGC. Place a book
recording the data. Have students take their
with a height of approximately 5 cm under the
stations.
front portion of the PGC.
0 Note: Have students rotate duties
so that each student
I. Have students select a wing shape to test.
can launch an aircraft.
Geometry and Algebra: The Future Flight Equation
EG-2001-09-30-LARC
1 0